INTENSITY OF GRAVITY.] 



ASTRONOMY. 



987 



Kg. 147. 



the earth's surface, the actual motion in either case 

 being in a curved line, A B. Let A B, on the orbit of 

 the planet, be the portion of that orbit described in one 

 second of time ; the distance D B will be that which the 

 planet has fallen, from its wonted path, towards the 

 sun, in this second of time, and the double of D B will 

 measure the attractive force of the sun at that distance. 

 Now, to find the length of D B we proceed as follows : 

 From the point B draw B E perpendicular to the 

 radius SA; draw also from B a straight line to the 

 point A' of the orbit diametrically opposite to the point 

 A, and make A E equal to D B. It is plain that the arc 

 A B differs insensibly from its chord, and may be re- 

 placed by it ; so that the angle A B A', being in a semi- 

 circle, is a right angle. In the right-angled triangle 

 A B A', we have the proportion A E : A B : : A B : AA' 

 (Euc. 8 of VI.), for B E cannot differ sensibly from 



parallelism with D A ; therefore, -^--H = ~A~A' " -^-E = 



A R2 A TO 2 



- As the arc AB is the portion of the orbit 



described by the planet in a second of time, it measures 

 the uniform velocity with 

 which it moves ; and we see 

 that the distance A E, or BD, 

 which the planet falls in that 

 time towards the sun, is 

 found by dividing the square 

 of the velocity by twice the radius of the orbit. We 

 have seen that the attractive force of the sun, acting at 

 that distance, is measured by double the distance fallen 

 in a second. The intensity of the force, therefore, at 

 the planet, is expressed by the quotient of the square of 

 its_yelocity by the radius of the planet's orbit. 



may now compare together the intensities of the 

 forces which act at different distances, or on different 

 planets, by means of the third law of Kepler. In order 

 to this, let us suppose that planets, moving uniformly 

 in circles round the sun, be situated at distances from 

 that central body proportional to the numbers 



1, 2, 3, 4, 5, Ac (A). 



To obtain the velocity of one of these planets in its 

 circular orbit, we must divide the length of the circum- 

 ference by the number of seconds occupied in describing 

 it ; and to get the square of the velocity, we must divide 

 the square of the circumference by the square of the 

 number of seconds. But the squares of circumferences 

 are as the squares of their radii ; that is, in the present 

 case, as the numbers 



1, 4, 9, 16, 25, <fec. 



Also, from the third law of Kepler, the squares of the 

 times of revolution are as the cubes of the radii of the 

 orbits ; that is, in the present case, as the numbers 

 1, 8, 27, 64, 125, Ac. 



The squares of the velocities of the several planets 

 will, therefore, be related to one another as the num- 

 bers 



4 9 16 25 



8* 27' 64* 125* 



11 

 3* V 6' 



& - 



(B). 



Or as the numbers 

 1 1 

 l > 2' 



And as the force acting on each planet is measured by 

 the square of its velocity divided by its distance, we 

 hall evidently discover the law by which the intensity 

 varies, from one planet to another, by dividing the seve- 

 ral numbers (B) by the corresponding numbers (A) ; 

 that is to say, the forces at the several distances (A) will 

 be as the numbers 



1 



1 



4' 9' 



1 ! 



lli' 25* 



or inversely as the square* of the distances (A). And in 



this way is the law of solar attraction established ; and 

 not only does this follow from the third law of Kepler, 

 but this law itself may be shown to be a necessary conse- 

 quence of the law of force just established. For, since 

 the length of orbit, divided by the uniform velocity, 

 gives the periodic time in seconds, and that circum- 

 ferences vary as their radii, the periodic time varies as 

 the radius divided by the velocity. Hence, calling the 

 periodic time P, the radius R, the velocity v, and the 

 force F, we have in symbols, 



*..-.*. 



V 2 



it 



If, then, F .-. /. oc R' .'. P 2 oc R3 ; so 





that if the force vary inversely as the square of the dis- 

 tance, and bodies move round the common centre of at- 

 traction in circles of different radii, the squares of the 

 periods of revolution will be to one another as the cubes 

 of the distances. 



Having proved that the relation observed by Kepler 

 between the periodic times and the distances is a neces- 

 sary consequence of the above-named law of gravitation, 

 Newton sought to determine whether the forms of the 

 planetary orbits were not also a necessary consequence 

 of the same law ; and he accordingly found that, under 

 the influence of that law of attraction, it was impossible 

 for a body to move in any other curve than a cunic sec- 

 tion ; that is to say, the orbit must be either an ellipse 

 (of which a circle is a particular case), a parabola, or an 

 hyperbola.* This may be proved as follows : 



Let v be the velocity in a circle of radius R ; ' the 

 velocity in a parabola, ellipse, or hyperbola, whose radius 

 of curvature at perihelion is K' ; then 



F " 2 = 1/2 - v n v " 

 ~R R"" K B ^ 



Now R' is what, in the doctrine of the conic sections, 

 is called the semi-parameter of the- curve ; and R being 

 the distance of the focus from the vertex of the curve, it 

 is also proved in that doctrine, that the curve will be 

 a parabola, an ellipse, or an hyperbola, according as 



v' 1 

 R' is = 2 R, .^ 2 R, or 7 2 R. Hence, if -^ = 2, the 



orbit is a parabola ; if "- Z 2 > the orbit is an ellipse ; 



<t'3 



and if - > 2, the orbit is an hyperbola. And as one or 



f 3 



other of these conditions must necessarily have place, 

 whatever be the velocity v', it follows that the planets 

 and comets must all move in one or other of these three 

 curves. 



The above established law of force is thus competent 

 to account for the revolutions of all the planets and 

 comets round the sun, as also for the motions of those 

 comets in the system, if any there be, which describe 

 parabolic or hyperbolic orbits, and which consequently 

 proceed onward in space continually, and never return. 



The same law of force accounts, in like manner, for 

 the revolutions of the satellites round the planets to 

 which they belong, for these all describe elliptic orbits. 



This general statement, however, is, in strictness, only 

 an approximation to the truth, although an approxima- 

 tion so close, as to be but very little at fault in explain- 

 ing the great phenomena of the planetary motions, and 

 therefore regarded by Kepler as strictly accurate. In 

 the foregoing reasonings the attracting body has been 

 supposed to be absolutely fixed in space. This is not 

 entirely consistent with Newton's laws of universal gravi- 

 tation, according to which all bodies in the universe 

 mutually attract one another with a force directly pro- 

 portional to their masses, as well as inversely propor- 

 tional to the squares of their distances. On account of 

 the masses of the planets being very small in comparison 

 with the mass of the sun, the rejection of the planet's 

 See and', Malktmatio, p. 608, el teq. 



